Claudius%20Ptolemy

1
Claudius Ptolemy

• Saving the Heavens

2
Euclids Elements at work

• Euclids Elements quickly became the standard
  text for teaching mathematics at the Museum at
  Alexandria.
• Philosophical questions about the world could now
  be attacked with exact mathematical reasoning.

3
Eratosthenes of Cyrene

• 276 - 194 BCE
• Born in Cyrene, in North Africa (now in Lybia).
• Studied at Platos Academy.
• Appointed Librarian at the Museum in Alexandria.

4
Beta

• Eratosthenes was prolific. He worked in many
  fields. He was a
• Poet
• Historian
• Mathematician
• Astronomer
• Geographer
• He was nicknamed Beta.
• Not the best at anything, but the second best at
  many things.

5
Eratosthenes Map

• He coined the word geography and drew one of
  the first maps of the world (above).

6
Using Euclid

• Eratosthenes made very clever use of a few scant
  observations, plus a theorem from Euclid to
  decide one of the great unanswered questions
  about the world.

7
His data

• Eratosthenes had heard that in the town of Syene
  (now Aswan) in the south of Egypt, at noon on the
  summer solstice (June 21 for us) the sun was
  directly overhead.
• I.e. A perfectly upright pole (a gnomon) cast no
  shadow.
• Or, one could look directly down in a well and
  see ones reflection.

8
His data, 2

• Based on reports from on a heavily travelled
  trade route, Eratosthenes calculated that
  Alexandria was 5000 stadia north of Syene.

5000 stadia
Syene
9
His data, 3

• Eratosthenes then measured the angle formed by
  the suns rays and the upright pole (gnomon) at
  noon at the solstice in Alexandria. (Noon marked
  by when the shadow is shortest.)
• The angle was 712.

10
Proposition I.29 from Euclid
A straight line falling on parallel straight
lines makes the alternate angles equal to one
another, the exterior angle equal to the interior
and opposite angle, and the interior angles on
the same side equal to two right angles.
11

• Eratosthenes reasoned that by I.29, the angle
  produced by the suns rays falling on the gnomon
  at Alexandria is equal to the angle between Syene
  and Alexandria at the centre of the Earth.

12
Calculating the size of the Earth

• The angle at the gnomon, a, was 712, therefore
  the angle at the centre of the Earth, ß, was is
  also 712 which is 1/50 of a complete circle.
• Therefore the circumference of the Earth had to
  be stadia 250,000 stadia.

712 x 50 360 50 x 5000 250,000
13
Eratosthenes working assumptions

• 1. The Sun is very far away, so any light coming
  from it can be regarded as traveling in parallel
  lines.
• 2. The Earth is a perfect sphere.
• 3. A vertical shaft or a gnomon extended
  downwards will pass directly through the center
  of the Earth.
• 4. Alexandria is directly north of Syene, or
  close enough for these purposes.

14
A slight correction

• Later Eratosthenes made a somewhat finer
  observation and calculation and concluded that
  the circumference was 252,000 stadia.
• So, how good was his estimate.
• It depends.

15
What, exactly, are stadia?

• Stadia are long measures of length in ancient
  times.
• A stade (singular of stadia) is the length of a
  stadium.
• And that was?

16
Stadium lengths

• In Greece the typical stadium was 185 metres.
• In Egypt, where Eratosthenes was, the stade unit
  was 157.5 metres.

17
Comparative figures
Stade Length Circumference Circumference
Stade Length In Stadia In km
157.5 m 250,000 39,375
157.5 m 252,000 39,690
185 m 250,000 46,250
185 m 252,000 46,620
Compared to the modern figure for polar
circumference of 39,942 km, Eratosthenes was off
by at worst 17 and at best by under 1.
18
An astounding achievement

• Eratosthenes showed that relatively simple
  mathematics was sufficient to determine answers
  to many of the perplexing questions about nature.

19
Hipparchus of Rhodes

• Hipparchus of Rhodes
• Became a famous astronomer in Alexandria.
• Around 150 BCE developed a new tool for measuring
  relative distances of the stars from each other
  by the visual angle between them.

20
The Table of Chords

• Hipparchus invented the table of chords, a list
  of the ratio of the size of the chord of a circle
  to its radius associated with the angle from the
  centre of the circle that spans the chord.
• The equivalent of the sine function in
  trigonometry.

21
Precession of the equinoxes

• Hipparchus also calculated that there is a very
  slow shift in the heavens that makes the solar
  year not quite match the siderial (star) year.
• This is called precession of the equinoxes. He
  noted that the equinoxes come slightly earlier
  every year.
• The entire cycle takes about 26,000 years to
  complete.
• Hipparchus was able to discover this shift and to
  calculate its duration accurately, but the
  ancients had no understanding what might be its
  cause.

22
The Problem of the Planets, again

• 300 years after Hipparchus, another astronomer
  uses his calculating devices to create a complete
  system of the heavens, accounting for the weird
  motions of the planets.
• Finally a system of geometric motions is devised
  to account for the positions of the planets in
  the sky mathematically.

23
Claudius Ptolemy

• Lived about 150 CE, and worked in Alexandria at
  the Museum.

24
Ptolemys Geography

• Like Eratosthenes, Ptolemy studied the Earth as
  well as the heavens.
• One of his major works was his Geography, one of
  the first realistic atlases of the known world.

25
The Almagest

• Ptolemys major work was his Mathematical
  Composition.
• In later years it was referred as The Greatest
  (Composition), in Greek, Megiste.
• When translated into Arabic it was called al
  Megiste.
• When the work was translated into Latin and later
  English, it was called The Almagest.

26
The Almagest, 2

• The Almagest attempts to do for astronomy what
  Euclid did for mathematics
• Start with stated assumptions.
• Use logic and established mathematical theorems
  to demonstrate further results.
• Make one coherent system
• It even had 13 books, like Euclid.

27
Euclid-like assumptions

• The heavens move spherically.
• The Earth is spherical.
• Earth is in the middle of the heavens.
• The Earth has the ratio of a point to the
  heavens.
• The Earth is immobile.

28
Plato versus Aristotle

• Euclids assumptions were about mathematical
  objects.
• Matters of definition.
• Platonic forms, idealized.
• Ptolemys assumptions were about the physical
  world.
• Matters of judgement and decision.
• Empirical assessments and common sense.

29
Ptolemys Universe

• The basic framework of Ptolemys view of the
  cosmos is the Empedocles two-sphere model
• Earth in the center, with the four elements.
• The celestial sphere at the outside, holding the
  fixed stars and making a complete revolution once
  a day.
• The seven wandering starsplanetswere deemed to
  be somewhere between the Earth and the celestial
  sphere.

30
The Eudoxus-Aristotle system for the Planets

• In the system of Eudoxus, extended by Aristotle,
  the planets were the visible dots embedded on
  nested rotating spherical shells, centered on the
  Earth.

31
The Eudoxus-Aristotle system for the Planets, 2

• The motions of the visible planet were the result
  of combinations of circular motions of the
  spherical shells.
• For Eudoxus, these may have just been geometric,
  i.e. abstract, paths.
• For Aristotle the spherical shells were real
  physical objects, made of the fifth element.

32
The Ptolemaic system

• Ptolemys system was purely geometric, like
  Eudoxus, with combinations of circular motions.
• But they did not involve spheres centered on the
  Earth.
• Instead they used a device that had been invented
  by Hipparchus 300 years before Epicycles and
  Deferents.

33
Epicycles and Deferents

• Ptolemys system for each planet involves a large
  (imaginary) circle around the Earth, called the
  deferent, on which revolves a smaller circle, the
  epicycle.
• The visible planet sits on the edge of the
  epicycle.
• Both deferent and epicycle revolve in the same
  direction.

34
Accounting for Retrograde Motion

• The combined motions of the deferent and epicycle
  make the planet appear to turn and go backwards
  against the fixed stars.

35
Saving the Appearances

• An explanation for the strange apparent motion of
  the planets as acceptable motions for perfect
  heavenly bodies.
• The planets do not start and stop and change
  their minds. They just go round in circles,
  eternally.

36
How did it fit the facts?

• The main problem with Eudoxus and Aristotles
  models was that they did not track that observed
  motions of the planets very well.
• Ptolemys was much better at putting the planet
  in the place where it is actually seen.

37
But only up to a point.

• Ptolemys basic model was better than anything
  before, but still planets deviated a lot from
  where his model said they should be.
• First solution
• Vary the relative sizes of epicycle, deferent,
  and rates of motion.

38
Second solution The Eccentric

• Another tack
• Move the centre of the deferent away from the
  Earth.
• The planet still goes around the epicycle and the
  epicycle goes around the deferent.

39
Third Solution The Equant Point

• The most complex solution was to define another
  centre for the deferent.
• The equant point was the same distance from the
  centre of the deferent as the Earth, but on the
  other side.

40
Third Solution The Equant Point, 2

• The epicycle maintained a constant distance from
  the physical centre of the deferent, while
  maintaining a constant angular motion around the
  equant point.

41
Ptolemys system worked

• Unlike other astronomers, Ptolemy actually could
  specify where in the sky a star or planet would
  appear throughout its cycle within acceptable
  limits.
• He saved the appearances.
• He produced an abstract, mathematical account
  that explained the sensible phenomena by
  reference to Platonic forms.

42
But did it make any sense?

• Ptolemy gave no reasons why the planets should
  turn about circles attached to circles in
  arbitrary positions in the sky.
• Despite its bizarre account, Ptolemys model
  remained the standard cosmological view for 1400
  years.